On an arithmetic convolution

Abstract

The Cauchy-type product of two arithmetic functions f and g on nonnegative in- tegers is defined by (f • g)(k):= Pk m=∑km=0 (km(m)g(k -m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.